Evaluate \( \sec^{-1}(\sec \frac{25\pi}{6}) \)
Step-by-Step Solution
We need to evaluate:
\[ \sec^{-1}\left(\sec \frac{25\pi}{6}\right) \]
Step 1: Reduce the angle
\[ \frac{25\pi}{6} = 4\pi + \frac{\pi}{6} \]
\[ \sec\left(\frac{25\pi}{6}\right) = \sec\left(\frac{\pi}{6}\right) \]
Step 2: Convert to cosine
\[ \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \Rightarrow \sec \frac{\pi}{6} = \frac{2}{\sqrt{3}} \]
Step 3: Apply inverse secant
\[ \sec^{-1}\left(\frac{2}{\sqrt{3}}\right) \]
The principal value range of \( \sec^{-1}x \) is:
\[ [0, \pi] \setminus \left\{\frac{\pi}{2}\right\} \]
Step 4: Find the correct angle
\[ \sec \theta = \frac{2}{\sqrt{3}} \Rightarrow \cos \theta = \frac{\sqrt{3}}{2} \]
In the interval \( [0, \pi] \), this occurs at:
\[ \theta = \frac{\pi}{6} \]
Final Answer
\[ \boxed{\frac{\pi}{6}} \]